66 CHAPTER 2 Sets To determine the ordered pairs in a Cartesian product, select the first element of set A and form an ordered pair with each element of set B. Then select the second element of set A and form an ordered pair with each element of set B. Continue in this manner until you have used each element of set A. Using set-builder notation, the difference of two sets A and B is indicated by − = A B ∈ x x A { | and ∉ x B}. The shaded region, region I, in Fig. 2.11 represents the difference of two sets A and B, or − A B. Notice in Fig. 2.11 that − = ′ A B A B. > U A B I II III IV A 2 B Figure 2.11 Example 11 The Difference of Two Sets Given abcdef ghijk b d e f g h a b d h i b e g U {,,, ,,,,,,,} A {, ,,,,} B {,,,,} C { , , } = = = = determine a) − A B b) − A C c) ′ − A B d) − ′ A C Solution a) − A B is the set of elements that are in set A but not set B. The elements that are in set A but not set B are e f , , and g. Therefore, − = A B e f g {, , }. b) − A C is the set of elements that are in set A but not set C. The elements that are in set A but not set C are d f , , and h. Therefore, − = A C d f h {, , }. c) To determine ′ − A B, we must first determine ′A. ′ = A a c i j k {, , , , } ′ − A B is the set of elements that are in set ′A but not set B. The elements that are in set ′A but not set B are c j , , and k. Therefore, ′ − = A B c j k {, , }. d) To determine − ′ A C , we must first determine ′C . ′ = C a c d f h i j k {, , , , , , , } − ′ A C is the set of elements that are in set A but not set ′C . The elements that are in set A but not set ′C are b e , , and g. Therefore, − ′ = A C b e g {, , }. 7 Now try Exercise 25 Next we discuss the Cartesian product. Cartesian Product Definition: Cartesian Product The Cartesian product of set A and set B, symbolized by × A B and read “A cross B,” is the set of all possible ordered pairs of the form a b ( , ), where ∈ a A and ∈ b B.
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